Math 8 Notes

Chapter 1

Plotting:

X is the horizontal axis

Y is the vertical axis

Example of an ordered pair

Ordered pairs are plots on a horizontal and vertical axis

A (5, 3)

The 5 is X and Y is 3

The 5 is the horizontal coordinate

The 3 is the vertical coordinate

Terms:

Mean: the sum of the data items divided by the total of data items. Example: 4, 6, 8, 10 the total for this set is 28 and the mean is 7.

Median: The middle number or the average of the two middle numbers when data is listed in order. 10, 20, 36, 40, 50, 60 - 38 is the median in this data set.

Mode: The most frequently occurring item in a data set. Can be one or more modes, or no modes. 5, 4, 3, 3, 5, 6, 7 – 5 and 3 are the modes in this set. 1, 2, 3, 4, - this set has no mode.

Range: the difference between the largest and smallest data items. For example, 12 is the biggest and 6 is the smallest, the range is 6.

Problem Solving from P. 9

I. Understand the Problem

A. Identify the question to answer

B. Find information to answer the question

II. Make A Plan

A. Pick a strategy to use in solving the problem

B. examples of plans: guess and check, act it out, make a
list . . . many more p. 9

III. Carry the Plan

A. If things go badly, pick a new plan

IV. Look Back (Grade the Plan)

A. Did your plan work? Why did it fail?

Rate: Ratio that compares two amounts measured in different units. Compares two quantities measured in different units.

$40 for 2 pairs of shoes

Unit Rate: Ratio that compares a quantity to one unit of another quantity. Gives an amount per 1 unit

$20 for 1 pair of shoes. Price per ounce. Price per pound. Inches of rain per year.

The order of operations is PEMDAS Using the order of operations and any 5 numbers between 1-19 will produce an answer between 21-49.

Equivalent Fractions

1mi ?mi 1mi x 5 5mi

3 hrs 15hrs 3 hrs x5 15hrs

6.3lb ?lbs 6.3lbs x 2 12.6lbs

1 hr 1hr 1hr x2 2hrs

6.3lb ?lbs 6.3lbs x 4 25.2lbs

1 hr 1hr 1hr x4 4hrs

Show your work to get the points. Rates and Unit Rates should look like the examples given above.

3.000 mph is an example of a unit rate.

Split Cherry Tree

Climax is when Luster sees germs through the microscope and he realizes the Professor Herbert is teaching Dave something of value. He realizes the future importance of education that his son will need.

Resolution. All the major characters undergo a change from the confrontation at school. Professor Herbert is willing to let Dave go without finishing working off the damages he did because he realizes that he is needed at home to do chores. He probably sees that some other punishment would have been more suitable for Dave. Dave realizes there is more to education than just what is taught in school. Even though his father is unschooled he has a lot of knowledge. Dave sees another side to his tough, stubborn father.

3,000 mph is a unit rate. It compares two different rates. Compares 1 quantity to another quantity.

Stem and Leaf Plots – Look like this

1 0 1 3

2 2 2 2 5 6

3 1 4 5 5 5 6

Scatter Plots

Can be used to compare things attempted to actual results. Fitted Line: splits the dots to show a pattern. Most common are straight line, curved line, and no pattern.

expressions

3m – 8 is 8 less than 3 x m

d/15 + 11 is 11 more than the quotient of d/15

Formulas for circles

circumference is the distance around a circle

radius is half the diameter of a circle

diameter is the distance across the center of the circle

r = d/2

Volume: l x w x h ( length times width times height)

Variable: symbol used to represent an unknown

Evaluate: Solve a mathematical expression

Exponent: using a power to multiply a number by itself a given number of times

Expression: mathematical expressions contain numbers, variables, and operations

Circumference is 3.14 times larger than the diameter. 22/7 is more accurate than 3.14.

Volume of a Rectangular Prism: V = l x w x h (length times width times height.

Volume of a Circular Cylinder: Bh (B times h) h is height. B is the base. The base is found by :

The base is like a roll of tape, and the height is like setting a can of pop on the roll of tape. The label for these problems is cubed. example inches ³

Writing Equations

Variables are chosen to represent unknown quantities. x is an example of a variable.

To solve an equation you must find a solution that makes the equation true. Inverse operations are used to solve equations.

Perimeter: The distance around the outside of an object. It is found by adding up the length of alhe sides.

Area of the Perimeter of a Polygon:

Polygon: A polygon is a closed plan figure with 3 or more segments that do not cross.

Regular Polygon: a polygon with all sides and angles congruent. (Congruent means having the same shape and size)

A polygon with a greater area than another polygon does not necessarily have a greater perimeter.

Area of triangle ABC is greater than DEF. DEF has a greater perimeter than triangle ABC.

Three ways to model a relationship between circumference and the area of a circle

1. Equations

2. Tables

3. Graphs

Module # 2

Surveys: Used to make predictions based on observations. Usually only one sample is used.

Estimating %

Make your math as simple as possible. Use a nice fraction to work with like changing 37/100 to 40/100. Multiples of 10% is the second way to find a fraction. Example 10% of 100 is 10 so by multiplying that by 5 we know that 50 is 50% of 100 too.

Solving percents

Use an equation

Example: 26% of 80 means .26 x 80 = 20.8

Use a proportion. Remember in using a proportion cross products are equal.

26 ₌ x

100 80

2080 = 100x x = 2080/100 x = 20.8

Sample Estimating %

34% of 4000 34/100 is close to 33/1000 which is 1/3

Thus 4000 / 3 is about 1300

Second Way of Estimating 34% of 4000

10% of 4000 = 400

20% of 4000 = 800

30% of 4000 = 1200

40% of 4000 = 1600

34% is close to 35% and 35% would be halfway between 1200 and 1600 so the answer is 1400

Exact answer 34% of 4000

34 ₌ x 136,000

100 4000 100 x x = 1360

Exact answer 37% of 1200

37 ₌ x 44,400

100 1200 100 x x = 444

Population: the entire group being studied

Sample: The part of the population being worked with

of = multiplication

is = equals

212

Examples

75% of the class goes to the mall

26 people in the class

.75 x 26 = 19.5 about 20 people

5% = 5 = .05

100

36 is 45% of what number. Remember that of means multiple and is means equal.

45%n = 36

.45 n = 36

.45 .45 n = 80

3 is what % of 14? Remember is means equal and of means mulitply.

3 = n14

3 = n14

14 14 n = .21 21%

Percent of Change

1. Percent of decrease

2. Percent of increase

Percent of Decrease – solved

Original Price – sale price = price decrease

Example 50 – 25 = 25 then 25/50 = .5 or 50%

Percent of Increase - solved

Increased price – original price = markup

Example 100 – 75 = 25 then 25/75 = .33 or 33%

Formula % of decrease = amount of decrease

original amount

Formula % of increase = amount of increase

original amount

Experiment: activity whose results can be observed and recorded.

Outcome: the result of an experiment. Example all pieces of the pie below

Equally Likely Outcomes: outcomes that have the same chance of occurring.

equal chance of landing on each section of the spinner.

Event: an event is the set of outcomes of an experiment. Example spinning and landing on one section is the event.

Probability: A number from 0 to 1 that tells how likely something is to happen. Theoretically probability is done without doing an experiment. A certain event is 1, and an impossible event is 0.

Dependent Events: The occurrence of one event effects the occurrence of another event.

Independent Events: The occurrence of one event does not effect the occurrence of another event.

Experimental Probability: found by conducting an experiment several times and recording the results

Tree Diagrams: used to show all the possible outcomes an experiment

More Examples of Percent of Change

1. Decrease. original $85 and the new $65

Step 1 $85 - $ 65 = $17

Step 2 $17/$85 = .2 = 20% decrease

2. Decrease. original $1.60 and the new $.95

Step 1 $1.60 - $ .95 = $.65

Step 2 $.65/$1.65 = .41 = 41% decrease

3. Increase original 126 and the new 150

Step 1 150 - 126 = 24

Step 2 24/126 = .19 = 19%

Theoretical Probability: # of outcomes that make up event

total # of outcomes

Outcome: result of an experiment

Equally Likely – outcomes that have the same chance of occurring

Event – set of outcomes of an experiment

Probability - # from 0 to 1 that tells how likely something is to happen

Impossible Event: an event that cannot occur

Certain Event: an event that will happen

Difference between theoretical and experimental probability. Take one dice, the theoretical probability of rolling a 6 is one out of 6. The experimental probability is the actual outcome. For example I rolled the dice 12 times and got 3 sixes. My outcome is 3/12 or 1/4 for rolling sixes.

Tree Diagrams: tree diagrams can be used to show all the possible outcomes of an experiment.

Adding & Subtracting Integers: The sum and difference of two integers may be positive (+), negative (-), or zero.

Examples: -3 + 5 = 2 12 + (- 12) = 0

Absolute Value: The absolute vale of a # tells you its distance from zero. Thus the absolute value of 4 is 4, and the absolute value of – 4 is 4.

Example of adding with the same sign (see the same sign box)

-16 + (-14) = -30. Just add 16 and 14 and then put the minus sign before your answer.

Example of adding with different signs (see the different sign box)

5 + (- 7) – 18 = -20 First combine the like terms of –18 and –7. You get – 25. Then subtract 5 from 25 to get 20 and put a minus sign in front of 20 because – 25 is bigger than 5.

Example of subtracting with the same signs (see subtraction problem box)

- 7 – (- 4) = - 7 + 4 ( see the different sign box) subtract these and put the sign of the larger number. –7 + 4 = - 3.

Example of subtraction with different signs. (see subtraction problem box first.

- 24 – (+4) = -24 + (-4) = - 28 (go to the same sign box) and add together.

Other examples

- 32 + 46 – (- 12) =

Step one –32 + 46 = 14

Step two 14 – (-12) see subtraction problem box thus 14 + 12 = 26.

- 17 – (-12) = -17 + (12) = - 5

- 17 – (+ 12) = -17 + (-12) = -29

Adding & Subtracting Integers

signs the same: Add and keep the sign

signs different: Subtract and keep the sign of the larger number

Multiplying and Dividing Integers

positive x positive = positive

positive x negative = negative

negative x negative = positive

positive divided by positive = positive

positive divided by negative = negative

negative divided by negative = positive

+	and	+	=	+	Positive when both sigs are same
+	and	(-)	=	(-)	Negative when signs are different
(-)	and